"I will take you, if you like, point by point along the line of reasoning which I followed myself, and which will inevitably lead you, as it led me, to the only possible solution of the mystery. "First take this point," he said with nervous restlessness, once more taking up his bit of string, and forming with each point raised a series of knots which would have shamed a navigating instructor, "obviously it was _impossible_ for Kershaw not to have been acquainted with Smethurst, since he was fully apprised of the latter's arrival in England by two letters. Now it was clear to me from the first that _no one_ could have written those two letters except Smethurst. You will argue that those letters were proved not to have been written by the man in the dock. Exactly. Remember, Kershaw was a careless man--he had lost both envelopes. To him they were insignificant. Now it was never _disproved_ that those letters were written by Smethurst." "But--" suggested Polly. "Wait a minute," he interrupted, while knot number two appeared upon the scene, "it was proved that six days after the murder, William Kershaw was alive, and visited the Torriani Hotel, where already he was known, and where he conveniently left a pocket-book behind, so that there should be no mistake as to his identity; but it was never questioned where Mr. Francis Smethurst, the millionaire, happened to spend that very same afternoon." "Surely, you don't mean?" gasped the girl. "One moment, please," he added triumphantly. "How did it come about that the landlord of the Torriani Hotel was brought into court at all? How did Sir Arthur Inglewood, or rather his client, know that William Kershaw had on those two memorable occasions visited the hotel, and that its landlord could bring such convincing evidence forward that would for ever exonerate the millionaire from the imputation of murder?" "Surely," I argued, "the usual means, the police--" "The police had kept the whole affair very dark until the arrest at the Hotel Cecil. They did not put into the papers the usual: 'If anyone happens to know of the whereabouts, etc. etc'. Had the landlord of that hotel heard of the disappearance of Kershaw through the usual channels, he would have put himself in communication with the police. Sir Arthur Inglewood produced him. How did Sir Arthur Inglewood come on his track?" "Surely, you don't mean?" "Point number four," he resumed imperturbably, "Mrs. Kershaw was never requested to produce a specimen of her husband's handwriting. Why? Because the police, clever as you say they are, never started on the right tack. They believed William Kershaw to have been murdered; they looked for William Kershaw.